We can treat temperate ice as a heat-conducting binary mixture of water and ice. The continuity equation for energy with the constitutive relations for internal energy = Cp + L and the nonadvective energy flux q = −k∇θ − νL∇W then yields
where D is the strain rate tensor, T′ is the deviatoric stress tensor, and tr denotes the mathematical operator trace. Variable ρ is the density of ice; ν is the moisture diffusivity, which is unknown. Equation (7) is not general since the extremely small contribution of kinetic energy is neglected, diffusion of heat is described as Fourier-type diffusion, which is realistic, and moisture diffusion is described as a Fickian-type diffusion, which is questionable. The temperature, θ, of temperate ice is assumed to be at the local pressure melting point of pure ice, and heat generated by mechanical or nonmechanical processes is immediately consumed by melting ice. Hydrostatic pressure, P, and the corresponding melting temperature can be written as a function of depth below ice surface, S. The hydrostatic pressure becomes P = Pr + ρg(S − z), and thus θ = θr − ϕ(P − Pr) = θr − ϕρg(S − z), where ϕ = 1.3 × 10−7 K Pa−1 is the pressure melting point depression, g is the acceleration of gravity, Pr and θr are a reference pressure and the corresponding reference melting temperature, respectively, and z is the vertical (parallel to gravity) coordinate. To rewrite equation (7) in the plane strain approximation (two-dimensional, following flow lines) in Cartesian coordinates (x, z), the corresponding temporal and spatial derivatives of temperature are substituted in equation (7):
where τxz and σxx are the shear stress and the longitudinal component of the deviatoric stress tensor, respectively, and u and w are the velocity components in the x and z direction, respectively. Note that the expressions for the thermal conduction and heat advection are not written in terms of temperature but in terms of surface geometry. The temperature is no longer an unknown field variable, and the corresponding expressions in equation (8) correspond to source terms for water content. With this, equation (8) is recovered as an advection-production equation for one unknown field variable, the water content W. To further simplify equation (8), heat and water diffusion are omitted [Blatter and Hutter, 1991], and steady state is assumed for the glacier geometry, moisture, and velocity fields:
Equation (9) is equivalent to
where WA and WB are the moisture contents at the starting and end points of the considered trajectory and integration is along the particle path. Q is heat produced by internal deformation. In our case the starting point A lies in the accumulation zone at a depth where water content is no longer affected by percolating melt water and the annual freezing front, and B is the position where the particle reaches the CTS. The initial value WA corresponds to the entrapment of water in the accumulation zone. The first and second term in the middle of equation (9) describe the water production by hydrostatic pressure changes and by strain melting, respectively. Hence by looking at the individual terms in equation (9), it is possible to get an estimate for the contribution of water content from strain heating and hydrostatic pressure changes. The contribution from entrapment of water at the ice-firn transition, WA, and the water input from the surface in the ablation area must be considered separately.